40 2. Background
it is negative another equation for the same plane may be obtained
by multiplying through by −1. It is not without loss of generality to
assume the coefficient of x is 1, since the plane defined by y + z = 0,
for example, has no equation with an x-coefficient of 1.
Exercise 2.6.1. Here are some proofs you can try that don’t involve
induction:
(i) ¬(∀m)(∀n)(3m + 5n = 12) (over N)
(ii) For any integer n, the number n2 + n + 1 is odd.
(iii) If every even natural number greater than 2 is the sum of two
primes, then every odd natural number greater than 5 is the
sum of three primes.
(iv) For nonempty sets A and B, A × B = B × A if and only if
A = B.
(v)

2 is irrational. (Hint: work by contradiction, assuming there
is a fraction of integers in least terms that equals

2.)
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